Theorem sseqtrrd 3287

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
sseqtrrd.1	|- ( ph -> A C_ B )
sseqtrrd.2	|- ( ph -> C = B )

Assertion

Ref	Expression
sseqtrrd	|- ( ph -> A C_ C )

Proof of Theorem sseqtrrd

Step	Hyp	Ref	Expression
1		sseqtrrd.1	. 2 |- ( ph -> A C_ B )
2		sseqtrrd.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2244	. 2 |- ( ph -> B = C )
4	1, 3	sseqtrd 3286	1 |- ( ph -> A C_ C )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   C_ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is used by:  sseqtrrid  3299  fnfvima  5953  tfrlemiubacc  6601  tfr1onlemubacc  6617  tfrcllemubacc  6630  rdgivallem  6652  nnnninf  7466  nninfwlpoimlemg  7515  ccatass  11390  swrdval2  11437  dfphi2  13018  ctinf  13370  imasaddfnlemg  13684  imasaddvallemg  13685  subsubm  13839  subsubg  14049  subsubrng  14571  subsubrg  14602  lidlss  14862  toponss  15176  ssntr  15272  iscnp3  15353  cnprcl2k  15356  tgcn  15358  tgcnp  15359  ssidcn  15360  cncnp  15380  txcnp  15421  imasnopn  15449  hmeontr  15463  blssec  15588  blssopn  15635  xmettx  15660  metcnp  15662  plyaddlem1  15897  plymullem1  15898  plycoeid3  15907  nnsf  17146  nninfsellemsuc  17153